Modern hullforms and parametric excitation of the roll motion

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prSHIEF

Technische H

EP. 1984

RCogeschool,

tab. v.

_{Scheepsbouwkunde}

Modern hullforms and parametric excitation of the

bit

motion

by

!are Lindemann, Principal Research _Engineer

Nere Skomedal, Research Engineer Det Norske Veritas, 1322 Hpvik, Oslo

Abstract: Modern hull forms with wide deck areas for better cargo handling procedures and sharp underwater lines to minimize resistance

is shown to be parametrically excited in roll motion (Mathieu instability).

A simple method to deduce the parameters of parametric excitation is introduced. The presented method is correct in beam sea, however,

extension to following sea shows good agreement with model experiments.

Model test is carried out with a vessel of typical modern hullform.

Parametrically excitated roll motion was observed in regular and irregular

beam seas as well as in irregular following _sea.

Using a simplified numerical simulation_{method approximately the same}

results and trends were obtained.

It is shown that ship fulfilling the minimum IMCO recommendations to stability may capsize, however, mounting bilge keels will improve the seakeeping quality considerably.

The mathematical formulation of the problem is discussed in some detail.

The influence of ship form, loading condition and roll damping devices are discussed.

Keywords: Ship hull, _{roll motion, parametric excitation, capsizing.}

I. INTRODUCTION

The seakeeping characteristics of most ships have_been

determined by linear theory, utilizing

_commonly

accepted strip theories such as the Salvesen, Tuck and Faltinsen (1970)[1] method.

The results, especially for motions in_{the vertical} plane, have been satisfactory in the _{sense that} com-puted motions have been confirmed by model

experi-ments and full-scale service experience even in relative

large waves. The same is true for the

_{roll motion}

provided certain linearization techniques have been introduced for roll damping.

However, modern hullforms designed forminimum

resistance and trying to attain full deck width over the entire ship length has not been underlaid _{the same} careful evaluation with respect to linearity in ship motions.

The R0/110 ship «FINNEAGLE»,

14.497tdw.

caught fire and was abandoned off the Orkney_Islands on 1 October1980. _{The fire was caused by a chemical} , reaction between acid from a car battery and liquid (trimetylphosphate) leaking froma damaged tank

con-tainer. The container broke loose in _{severe weather.}

K. Lindemann N. Skomedal

From the report recently issued by the Board

of

Investigation looking into the accident we find that

the ship was sailing in following_{seas at reduced speed}

when it suddenly experienced three severe overturns (roll cycles), listing up to 40°. This probably caused

the cargo to break loose.

From linear or linearized roll theory roll motion _of this nature could not occur under such _conditions. This indicates that the sudden roll motion experienced was caused by a highly nonlinear phenomena.

In this work the roll motion caused by the time dependant variation of ship's transverse stability in a seaway will be studied in some detail. As a result of this time dependent righting arm variation, an unstable resonant roll motion may suddenly develop when a ship encounters waves of certain frequencies. This is most likely to happen in following seas and_{as such be}

a possible explanation for the severe roll motion

experienced by «FINNEAGLED.

This unstable motion is termed parametric excita-tion and was firstly studied by Grim(1952) [2], who

considered the case of regular sinusoidal

_waves.

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periodic waves has subsequently been studied by Ker-win (1953) [3], Paulling and Rosenberg (1959) [4], Abicht (1975) [5], Kure and Bang (1975) [6], Blocki (1980) [71, Feat and Jones (1981) [8].

Paulling et al. (1972) [9] tested a model in San Francisco Bay in order to study capsizing and control performance of the model in heavy seas.

All the capsizes occurred in following or quartening seas. It was possible to distinguish three modes of

capsizing.

Broaching

Pure loss of stability Low cycle resonance

The latter mode, which is most interesting in the point of view of this report, occurred as the model encoun-tered a group of large or breaking waves. The waves seemed to encounter the ship at nearly the double of natural roll frequency. In some cases the encounter

frequency occurred at the natural roll frequency.

These are the frequencies where the Mathieu differ-ential equation has it's unstable regions. (The Mathieu differential equation is used to model the nonlinear roll motion discussed in this paper).

In the later years some attempts have been made

to study the problem of parametric excitation in irregu-lar sea utilizing stochastic process theory. Contributors

have been Price (1975) [10], Haddara (1975) [11] and recently Muhuri (1980) [12] and Roberts (1980) [13]. It will be shown in this paper that parametric

exci-tation of the roll motion may occur, even in an irregular

and moderate seaway for certain hullforms.

It is worthwhile to notice that most papers on par-ametric excitation until now have applied a harmonic variation in the parametric excitation. In this analysis nonlinear parametric excitation will be taken into account. This can easily be done in the presented simulation procedure.

2. THEORY

The roll equation for a ship in a seaway may generally be written as:

A(t)4). + B(t)0 + C(t)(p

+ D(t) cp + E(00 = F(t) (1)

where following assumptions are made:

A(t) represents - the massmoment element of roll motion.

B(t) represents the dissipative element, the roll damp-ing and is regarded as a function of pure roll

velocity.

C(t) represents the

total

restoring element and

includes the time dependent variation of meta-centric height, variation of displacement force

3

and non-linearities in righting arm as function of

heel.

F(t) represents the roll exciting moment.

D(t) represents the coupling from sway into roll. E(t) represents the coupling from yaw into roll. The two latter elements have for simplicity been omit-ted in the present analysis. .

An uncoupled roll equation may be expressed as

follows:

+ A(0)0+ B

+ gA (t)GM(00 = F(t) (2)

assuming that A(t) is constant and the subscript o indicates values at still water level, we have:

(I + A)+ B(00 + g A. GM°

- (1 + h(t)) = F(t) (3)

where

h(t) = (A(t)GM(t)/AGM,,) 1 (4)

and is termed the Mathieu parameter. On a

non-dimensional form eq. (3) become:

+ 2p(t)w0

+ w(1 + h(t))

= f(t) (5)

By neglecting the exciting force, f, and assuming that h(t) is a harmonic function and that the dissipating term is constant, we will after applying the following transformations:

774 = y(t)e-Pale.`, y(t) = y sin wt, T= We t

obtain the classical Mathieu equation:

d2y/dr2 + (coo/ coe)2((1 p2) + h cos r)y = 0 or on a standard form

d2y/d r2 + (5+ e cos r) y 0 (6)

with

6 = (coo/ we)2(1 p2), and the tuning factor

E= h(co0/co.)2

where h represents the variation in the restoring force due to the ships motions in a seaway, e.g. the time dependent variation in displacement and GM.

The general solution of the Mathieu equation is not known, however, its stable and unstable regions can

be deduced, as shown in Fig. 1 [14] with zero damping.

The stability chart shows that unstable regions

approach the 6-axis near

6 = n2/ 4(n = 1, 2, 3, ...)

The most interesting occurs near 6= 1/4, because in this band the unstable region is quite broad for

small values of E.

So far the analysis has followed a classical approach.

Norwegian Maritime Research No. 2/1983

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STaast *newt SeACCD

Fig. I. Stable and unstable regions for the Mathieu

equation.

This has been done to demonstrate the well known nature of eq. (6), with its stable and unstable regions. It is reasonable to assume that eq. (2) could be of the same nature as eq. (6) with the same stable and unstable solutions, also when p(t) varies with time and h(t) is a nonharmonic function. Time simulations of eq. (2) with a nonharmonic behaviour of the restoring and dissipating terms have been carried out and dem-onstrated the correctness of this assumptions, Sko-medal (1982), [15].

3.. THE METHOD OF SIMULATION

The roll equation is simulated using the general simu-lation package «Supersimx). This is done on a NORD

100 computer using a Runge-Kutta simulation method.

(Krogh 1975) [17].

The vessels are given an initial

roll angle at

time = 0. The initial value has no or negligible influ-ence on the result when the roll motion is undoubted stable or unstable. In the labile cases, near the borders

of the stable/unstable domains the effect is

more

significant.

The mathematical model of the roll motion used in the numerical simulation is as follows:

+ A)4) + BL(1 + Bc02)4)

+ A. GM0g

(GZ( (P)+h(t)) = F(t) ₍₇₎

Gli/100

where GZ( 0) is represented by a table and F(t) is for the case of regular beam sea written as

F(t)= co2 - GM. cos(cot Jr/2)

4. THE RESTORING MOMENT (MA'! HIEU

PARAMETER)

The time dependent variation of the Mathieu par-ameter, h, is due to coupling from vertical motions,

primarily heave, pitch and wave profile along the ship.

Paulling and Rosenberg (1959) [4], Kure and Bang (1975) [6] expressed the heave-roll coupling as

Kzo = 2p, 1.FPy2(x)dy(x)dx

.AP dz

+pswg - awl, (8)

and coupling from pitch to roll:

= 2pswg

.1"xy2(x)cl:(x)dx

AP UZ

PswgZWL aWL XF (9)

where the terms were derived based on a Taylor

expansion of the time dependent coefficients.

x is parallel to longitudinal axis y is the transverse axis.

The expressions (8) and (9) which are directly pro-portional to the Mathieu parameter h, are convenient

to use in order to demonstrate the design factors

having influence on the Mathieu parameter, e.g.

pos-sible instability. It is seen that h is proportional to

dy/dz, e.g. the slope of the hull side close to the

waterplane. A large slope will cause large values of the Mathieu parameter and an increased probability for having a parametric excited roll motion. Hence attention should be given to such hullforms, since the heave induced roll motion may cause an undesirable large roll motion. Traditional hullforms have nearly vertical ship sides amidship and some slope near bow and stern.

Ship hulls with barge formed sterns and large flare have considerably larger coupling coefficients. These hullforms, which will be termed untraditional hull-forms in the following, is often seen on container vessels, ro-ro ships, ferries and in the recent years on so-called «energy economical hullformso. Figure 2. Such hulls are more likely to be exposed to parametric excitation than traditional hullforms.

In the following we have limited our discussion to the heave-roll coupling since this is believed to be the

-most dominating factor causing parametric excited roll

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Fig. 2. Difference between untraditional hull form (to left) and traditional hullfonn (to right).

case the vertical relative motion of the ship will cause variation in the instantaneous displacement and GM. As a further approximation, the relative motion of the ship amidship will be used over the entire ship length. This is believed to be a fairly good approxi-mation in the case of longer waves, or in beam seas for all practical experienced waves.

The time dependence of the relative motion may be considered as a variation with draught. The Mathieu parameter may then be expressed as:

= = (d) GM(d)

h(t) h(RM) h(d) A.Gm.

and GMO are initial values and A(d) and GM(d) are instantaneous values.

h(t) may also be looked upon as a measure of the amount of energy which possibly may be transferred from heave to roll, giving rise to roll motion.

E (or h) is plotted for a number of load conditions and different hullforms in Figs. 3-5 at (co0/G.)2 = 1/4. From these figures it is seen that the &parameter varies with draught, in this example the sensitivity to parametric excitation decreases with draught. In the same way it may be shown that the Mathieu parameter also varies with trim. As seen, different hullforms will also expose large individual variations in the Mathieu parameter.

In addition to the hull shape, the ships load condition will influence on its sensitivity to parametric excitation.

As seen from eq. (4) a small initial GM will give rise to a large Mathieu parameter, while a large GM o tend to reduce its magnitude. Hence for a ship with a hull-form which is sensitive to parametric excitated roll motion, it is most desirable to operate the ship with a relative large GM..

The ship is most likely to be parametrically excited in roll when the wave period of encounter is about half that of the natural period of roll. Hullforms with a roll period about twice the wave-period for stormy

seas, may give a possible rise to a severe heave induced

roll motion. This is the case since the relative motion

5

Fig. 3. The &parameter plotted as a function of relative

vertical motion for a number of ships. GM01 B Ls constant and relative high. All vessels have a small e-parameter resulting in negligible par-ametric excitation of roll motion.

A

\

E MIMEO -MAWR 1.0 0.60 0.20 .00°-"

.

-0.20 -0.40 644 04186

N.

4 ( m) RELA14 VERT. MOTION

Fig. 4. The e-parameter plotted as a function of relative

vertical motion. GM0 has relative low values. A, B, C, and D are untraditional hull forms

where D is used in the model experiments. E is a traditional hullform. The large difference between individudl hullforms indicate different roll properties.

Norwegian Maritime Research No. 211983

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Fig. 5. The influence of draught on the &parameter,

GM01B is constant.

at these wave frequencies will be of such a magnitude that substantial amounts of energy may be transferred into roll.

The GM sets the ships natural roll period. The roll period is inversely proportional to the square root of GM. An increase in GM will reduce the natural roll

period. Hence an increase in GM to reduce the

non-linear energy transfer from heave into roll, may

cause the resonance frequency to shift to periods with

significantly more wave-energy. This in turn may cause

a more severe linear roll motion.

As seen, the restoring moment has considerable

influence on the nature of roll motion. The time

dependency will, if excited at the right frequencies cause the ship to roll in resonance, even when no wave exciting moment is present. Hull lines and metacentric height was shown to be the two dominating factors influencing on this behaviour.

5. THE DISSIPATING MOMENT (DAMPING) Returning to eq. (2) Kure and Bang (1975), [6] ana-lyzed the Mathieu equation with a linear damping moment. They applied an energy balance analysis and obtained a stability criteria at 6= 1/4 as:

The roll motion is stable when p E.

In this case, the e value may be termed the threshold value.

In Fig. 6 we have simulated the Mathieu equation

applied by Kure and Bang (1975), [6] over a large range of the 5-axis. It is seen that by introducing a damping term the instability region is decreased. A larger Mathieu parameter is needed in order to have an instability. In practice this means that in order to be parametrically excited in roll the relative motion have to increase, or that a Mathieu instable ship may be made stable by increasing the roll damping.

In order to obtain a more representative

math-ematical model of the roll motion, non-linear damping should be included.

An often used model is linear plus a quadratic

damping:

BO= 2/x0044 + C0101 (10)

However, in several cases, it has been seen that a linear plus cubic damping gives a better fit to experi-mental data, Dalzell (1978), [16]:

B = 2pw0(I + A)

+ pc/o(1+ A) 03

(11)

In our general simulation model we have applied the latter expression.

6. NON-HARMONIC PARAMETRIC EXCITATION

Earlier works on parametric excitation of roll motion have mostly included the exciting parameter h as a sinusoidal harmonic function.

In the following an assumption with sinusoidal

rela-tive vertical motion resulting in a non-linear and

non-sinusoidal variation of the metacentric height will be introduced.

In order to simplify, the relative motion amidship will be used, and the parameter h may be calculated

as shown in eq. 4. This assumption is valid in following

and head sea when the wavelength is considerably larger than shiplength, L. In beam seas, the assump-tion is correct.

A table of h-values as a function of vertical relative motion may easily be calculated. This non-harmonic description of the Mathieu parameter has a significant influence on the roll motion. Simulation done with a

harmonic and a non-harmonic variation in the Mathieu

parameters has shown a ship to change from an

unstable condition with harmonic variation to a stable

with a non-harmonic variation

in

the Mathieu

parameter.

In this connection it should be noted that the

non-harmonic variation in the restoring moment will cause a shift in the roll resonance period as well (usu-ally to smaller values).

7. PARAMETRICALLY EXCITED ROLL MOTION IN IRREGULAR SEA STATES

The applied numerical simulation technique may be extended to irregular sea states in the followingway:

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0, 20 0,15 0,10 0,05 , 2 0,15 0,20 0,25 0,30 0,35 0,40 _,=, We _,Lf

Fig. 6. Stability chart of the linear Mathieu equation near =0.25. Stability borders with damping found from numerical simulation.

From wave spectrum and transfer function of rela-tive vertical motion amidship the response spectrum is calculated according to

S(w) = TRw(w)2* Sw(w) (12)

Dividing the response spectrum into a certain num-ber of intervals, the time series of relative motion is written as:

RM(t) =

am cos(ak t

Ei) (13)

7

where

N is number of intervals

aRmi is amplitude of relative motion aRmi = V2S1m(riA)

= random phase Aro = frequency interval

= frequency within interval i.

The time serie of variation of the Mathieu parameter is then calculated from a table of h-parameter as func-tion of relative mofunc-tion.

p.0,05

P =0,04 P = 0,03

P -0,02 p-0,01

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;?1_1-Further extension of this method may easily be done, introducing a number of stations were relative motion and variation of the Mathieu parameter for each station are calculated giving as a result the total variation in the Mathieu parameter at each instanta-neous point of time.

8. MODEL TEST

In order to control the validity of the proposed math-ematical simulation model, model experiments were carried out at the Danish Ship Research Laboratory. The model used in the experiments had dimensions as shown in table 1 and lines as in Fig. 26.

Table 1. Model data.

The lines are typical for modern hullforms, often seen on Ro-Ro, container and similar vessels.

The model was fitted with two sizes of bilge keels. See table 2.

BBK/B LBKIL, p A ian[s-')

Table 2. Bilge keel and damping data. Model experi-ment results. GM01 B = 0,0186 Natural roll frequency referred to fulLscale. (Parameters of damping as used in eq. (11))

For all

six degrees of freedom motions were

recorded using a system of a light triangular tower, thin threads and precision potentiometers.

The model was suspended in a system of weak

rubber springs. The suspension system had natural periods outside range of natural periods of roll or wave excitation.

The ship was free to move due to second order

driftforces in longitudinal direction of the model tank. The relative motion amidship was of great interest in this test set-up. This motion was derived from wave

probes positioned on each side of the model at

amidship.

As seen from test records from regular beam seas

some higher harmonic components (2. and 3.) in the relative vertical motion response occurred.

This effect is to the authors' knowledge not reported before, however, we believe that the effect is mainly

caused by non-linearities due to large changes in

waterplane area as function of draught.

This non-linearity is not seen on the heave response record or in its Fourier transformed signal, however, relative vertical motion is the difference between two input signals, the wave and heave. If those signals are slightly nonsinusoidal and at the same order of mag-nitude, the relative motion may include a relative high degree of higher harmonic components.

On the other hand, higher components of the out-going wave potential due to the vertical response can-not be excluded. Other effects as shadow effects and spray, is to our opinion of minor importance.

The desired wavespectra were of Pierson-Moskow-itz type, generated by a pseudo-random procedure. The waves were generated by means of a papertape controlled pneumatic wavemaker.

9. A COMPARISON OF MODEL EXPERIMENTS

AND NUMERICAL SIMULATIONS

The applied numerical simulation model is a very simplified mathematical model of the real situation, as the coupling effects from sway and yaw are

neg-lected, the existing wave induced roll moment is

excluded in irregular seaway and finally only the rela-tive motion amidship is taken into account. Despite these factors the obtained results compare very well

to the model experiments. A brief comparison of

model and simulation results is shown in table 3.

In this connection it should be mentioned that the roll response utilizing traditional strip theory [1] and short term wave statistics, gave in a sea state with a significant waveheight of 10 meter, a roll angle of 5 6 degrees. This is approximately the same value as obtained in the model experiments carried out under

the same conditions. However, the mathematical

simulation gave no roll angles, a consequence of not

including the linear roll exciting moment in the model.

In this case the model was not parametrically excited in roll. Figs. 13 and 14.

On figure 8 the results from model test in regular beam sea with medium bilge keels is shown. The tuning factor a = 0,28. The roll response is undoubted stable even if the model is given disturbances of 6-7

degrees. The corresponding numerically simulated roll

response is in good agreement and is shown in Fig. 9.

In figure 10 the tuning factor 5= 0,21, and the

waveheight is increased, resulting in a nearly doubling of the relative vertical motion. In this case the roll motion is unstable reaching a maximum value of 10

12 degrees.

The simulation shows the same picture (Fig. 11).

No bilge keels 0,014 2,99 .1970

Medium bilge keels 0,00976 0,247 0,026 6,56 .1933

Large bilge keels 0,0190 0,494 0,045 24,2 .1870

Length of model 5,58 m

LppiB 5,3

B/d 4,0

KG/B 0.47

GM/B 0.0186

Radius of inertia, roWB 0,38

Radius of inertia, pitch/L,, 0,31

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Relative vert. motion Roll motion

Values within parenthesis represent the Fourier component at wave frequency. t Reference is made to Fig. 19, largest amplitude before capsize: ca. 33 deg.

Significant higher components of relative vertical motion frequency observed.

Table 3. Comparison of model experiments andnumerical simulation.

In irregular beam sea, as shown in Fig. 12 and 13 no parametrically excited motion occurs. The sameis observed in the following figures with large bilge keels

(Fig. 14 and 15).

In Figs (16 and 17), the model (with large bilge keels) is in following sea at a speed of 3-4 knots. Parametric excited roll motion occurs, however, it's magnitude are relative small (-50).

In Figs 18 and 19 the bilge keels are removed. The

model is in following sea, speed approximately 5 knots.

In the first 7-8 minutes (fullscale time), Fig. 18, of the test record nothing happens to the roll motion.

Sud-denly after 8 minutes, (Fig. 19), the roll angles increase finally resulting in a capsize to port, after 10.5 minutes with severe rolling. (Time is reset to zero in the figure).

The corresponding numerical simulation is shown in

Fig. 20.

From the time history of the wave, it is seen that

a number of large waves of the same period and height

resulted in large values of vertical relative motion. This caused an extensive energy transfer from heave into roll and finally the model capsized. (The model did not fully comply to the IMCO stability recom-mendations in this case (see Fig. 7)).

It should be noted how sudden the roll motion

increase when first initiated, nearly without any

warning.

In an earlier model test with the same model,

with-out bilge keels, parametric excited roll motion

occurred even in beam sea. See Fig. 21.

9

Fig. 7. GZ-curves, GM0 = 0.0186.

- GZ-curve of

modelled ship. -: IMCO-minimum curve of

stability.

Norwegian Maritime Research No. 211983 Seastate reg/irreg. H/1-11/3 TA; Amplitude/ sign. value Exp. Sim. T , Exp. Sim. Largest amplitude Exp. Sirn. T. Exp. Sim. [m] [m] [m] [m] [s] [s] [deg] [deg] [s] [s] 6.0 17.3 0.4 0.23 17.3# 17.3 Stable roll 34.2 33.5 (0.21) motion (after disturbancy) 7.0 15.0 0.75 0.37 15.0# 15.0 12.0 12.2 30.8 30.0 (0.47) 8.7 11.3 1.66 1.68 8.2 8.4 Stable roll motion 27.1 29.9 (No parametric excitation) 10.1 12.2 1.66 1.28 83 8.2 Stable roll motion (No parametric excitation) 8.9 14.4 2.07 1.67 12.6 10.7 4.4 4.6 30.0 30.9 8.3 13.4 1.98 1.71 11.6 9.78 Capsizet 30.0 26.4 26.7 Run Comment

1 Reg. beam sea, medium

bilge keels

3 Reg. beam sea, medium

bilge keels 12 Irreg. beam sea

medium bilge keels

21 Irreg. beam sea, large bilge keels

22 Irreg. following sea, large bilge keels 23 Irreg. following sea,

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-5.00.. ROLL 1:115/0 '11 ' 11,1"11 11,11111.";.11111111111111 'I II TIME DI MUTER 4 .00- HEAVE CIO TD IE Dr MUTES TM III MINUTES PITCH COIN ME DI

Mini

Fig. 8. Time history of model experiments. Regular beam seas, H = 6.0m, T = 17.3 sec. Medium bilge keels. The arrows show where the roll motion were given a disturbancy.

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(DEG) ' ROLL

6 AMFUTUDE RUN NO. 1

MEDIU1 ELGEKEELS 4 -2 0,0 4 100 300 500 700(s)

Fig. 9. Numerical simulation of run shown in Fig. 8. The roll motion dies out.

(DEG.), ROLL

15 AMPLITUDE RUN NO. 3

MEDLII BILGEKEELS

5 --10

700 3.00 500 700 ( s

Fig. 11. Numerical simulation of run shown in Fig. 10.

Parametric excitation of the roll motion is observed. A stable level is reached where exciting work equals damping work.

The roll motion is stable.

11 (DEG.)' ROLL AWISILOE 4 2 0,0 2 -4 Seastate i;) 1134 NO. 21 LARGE BILGEKEELS 100 5100 700 (5 t 3

The roll motion is stable.

In the table the result from a model test is compared to a numerical simulation:

Model test largest roll

ampli-tude during run

However, a numerical simulation, of the same model in following seas without bilge keels and complying with all IMCO recommendations has been carried out, (Hs = 10 m). As seen in Fig. 22, the vessel capsized

as well.

The next two Figs (23 and 24) demonstrate the influence of roll damping on the roll motion. It is seen

(DEG.)1 ROLL RUN NO. 22

AMFUTILIE LARGE BILEBEELS

Simulation largest roll

ampli-tude during run

Runtime: 600s

100 300 9:10 700 (s)

Parametric excitation

of roll

motion is

observed. The motion is finite.

Norwegian Maritime Research No. 2/1983 Beam sea = 8,93 m = 10,16 5 24,1 deg 21,9 deg P-M-wave spectrum 10 5 0,0 1

LIII

10-5 0,0

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JAL ,L1LLI AilLJ

15.0

Tild 11410/2TJ

45.0

4$.9

Fig. 10. Time history of model experiment. Regular beam sea, H = 7.0 m, T = 15.0 sec. Medium bilge keels. Parametric excitation of the roll motion is observed.

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iii/ I .1 ! N.M. ANIDOMIP CM) 5.04 UAVE SIDE CM) 0.9 _-9.0 SO

TIME IN MINUTES _{TIRE IN}

mutes

TIME IN MINUTES TIME IN MINUTES

Fig. 12. Time history of model experiments. Irregular beam seas, H, = 8.7m, T,

= 11.3 sec. Medium bilge keels. No parametric

excitation of roll is observed.

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-8.00 4.89-YAW" CDEO %Le 0.80 11.11 I II ' ''

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TIME. IffPLINUTES TIft

usnutne3 5e .. I Iii 11,14111H-1 : TIME IMIUNUT03

Fig. 14. Time history of model experiments. Irregular beam

seas, H, = 10.1 m, T,

12.2 sec. Large bilge keels. No parametric

excited roll motion is observed.

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VII WI

10.0 TIME IN MINUTES

Fig. 16. Time history of model experiments. Irregular following

seas, H, = 8.9 m, T, = 14.4 sec. for spectrum of encounter. Ship

speed is 4 knots. Large bilge keels. The roll motion is

parametric excited, but finitie.

19.0 0.0

5.00-.M. AMIO3NIP CM)

1

(15)

4.60 ROLL CDEGN 0.06 -4.80 e ea 00 -8.9_ TIME IN MINUTES

Fig. 18. Time history of model experiments. Irregular following seas, H = 8.5 m, T, = 13.6 sec. Ship speed 5 knots. No bilge keels. Parametric excited roll motion is observed at time 8 minutes.

4.00_ HEAVE CM) 0 co -4.00

fiv

fiNx

4

/NINA

foi

TIME IN MINUTES

4.80 R.N. AMID$HIP CM) TIME IN MINUTES

(16)

se

i0

0 ROLL CDEO)

I Ali

AA Ai

LA

AAA

.a 7Y11

v v1 Y" VT

v1

VAVE SIDE CN)

Mil

TIME IN MINUTES TIME IN IlDiUTES 4. 99- AKIDSHIP CIO

TIME Iii nzmurce

TIME IN MINUTES

-2.40...

TIME IN MINUTES

Fig. 19. Continued time history of model experiments shown in Fig. 18. Time scale is reset to zero. Continued parametric excitation of roll motion. Capsize to port at time 11 minutes. (Some errors in the sway

response are due to problems with the measuring gear at such large roll amplitudes.)

17

(17)

Fig. 20. Numerical simulation of run shown in Figs 18

and 19. The dotted line show the relative

verti-cal motion with arbitrary dimensions. Severe parametric excited roll motion is observed. that a ship which may capsize in a seaway may obtain a good roll behaviour with no chance to capsize, if the

roll damping is increased sufficiently. Exactly the same

30 20 10 0 - 10 30 20 10 -10 10 -- WAVE AZIRITUIDE 5 $ -5 10 2S0 ROLL AMFLITLOE JJ RUN NO. 23 NO BI.EELS 300 350 400 450 500 550 600 (s )

Fig. 21. Time history of model experiments. Irregular beam seas, H, = 8.9 m, = 10.1 sec. No bilge keels. _{Parametric excited}

_{roll motion}

_is observed. SO 60 30 0,0 -30

Fig. 22. Numerical simulation of model with a

GZ-curve complying

with IMCO minimum

requirements. Irregular following seas, H, =

10.0 m, 7', = 12.0 sec. No bilge keels. Capsize after 568 seconds.

time history of wave amplitude

_{was used in the}

simulation.

The roll motion of some of the vessels for which the E-parameters are plotted in Fig. 4 is simulated and the response shown in Fig. 25. It is seen that the lines have a large influence on the roll motion. Vessel B has a large non-linear &parameter as function of verti-cal relative motion, and vessel C has a relative small slope. Both reach a constant maximum roll angle if waveheights are increased beyond a threshold value. It was not possible to obtain parametric excited roll motion of the vessel E (traditional hullform).

The &parameter function of vessel A is nearly linear

and has a relative steep slope. Large roll amplitudes at extreme waveheights are observed.

10. PRACTICAL APPLICATION IN DESIGN AND

OPERATION

In this paper the importance of parametric excited roll motion on modern hull forms has been demonstrated.

(MA Rai. ItICO- MIN

A/V(17XE til331111 BILGEKEELS

20 10 0,0 -10 -20 I I 100 2130 300. 400

Fig. 23. Numerical simulation of model with a

GZ-curve complying

with IMCO minimum

requirements. Same sea state and heading as in Fig. 22. Medium bilge keels. Substantial

parametric excited roll motion, but no capsize.

930 (s)

25 50 715 100 1'25 1;0 175 ( s

REGULAR BEAM SEA

NO BILGEKEELS

ROLL

AMPJTUDE

(18)

( DEG ) A 40-30 20 10 73 ROLL 0.0 -25 - 5.0 -10 200 3100 4100 500 s

Fig. 24. Numerical simulation of model with GZ-curve

complying with IMCO minimum require-ments. Same sea state and heading as in Fig.

22. Large bilge keels. Modest parametric

excited roll motion, no capsize.

ROLL AMPLITUDE LARGEST Rai. ANGLE DURING SIMULATION RUN. (Ca. 8 )

,/ A

IMCO MIN LARGE BILGEEELS

It is established that for such ships, complying with IMCO requirements for stability, severe roll motion

may occur in a relative moderate seaway, which should

be of concern to ship designers and operators. However, simple means exist to control the

likeli-hood for a design or a ship to be exposed to parametric

excited roll motion.

From hydrostatic tables, the &curves may simply be computed according to the following equation:

E(d) = 1/4(6.(d)GM(d) 1)

A0GM.

If this parameter at d draught ± 2 meter is

sig-It

i,

ec

WAV9fIGTH

5 10 15 H ( m)

Fig. 25. Numerical simulation in a regular beam sea

of different models with &parameter as shown

in Fig. 4. The wave period is half the natural roll period. Parametric excited roll motion is observed at certain wave heights. An equilib-rium condition is attained for ships B and C.

19

nificantly larger than the linearized damping p, the ship is likely to be exposed to parametrically excited roll motion. For a further discussion, the reader is referred to [15]. The linearized damping may vary considerably depending on the lines and choice of damping devices (bilge keels). No general guidelines as to the selection of this parameter may be given, but as a rule of the thumb we suggest that a value of 0.03 is chosen.

If this initial control indicates that the ship may be subjected to parametric excitated roll motion, a closer

study of the ship's damping characteristics is advisable.

However, as previously demonstrated, the roll

damping has a large influence on the parametric

excited roll motion. Hence, a simple mean to reduce or even eliminate the effect of parametric excited roll motion will be to increase the roll damping. This is most simply done by increasing the area and extent of the bilge keels.

On the design stage it may also be possible to alter the lines of the hull itself, either by giving the ship sharper corners or reducing the relative change in water plane area with draught.

The area and extent of the bilge keels has certain practical limitations. Based on our experience, the

needed size of a bilge keel to reduce the problem

associated with parametrically excited roll motion

should not be

in

conflict with maximum size

limitations.

In an operative mode of the ship, the influence of parametric excited roll motion may be significantly reduced by changing the load condition. Two

possi-bilities exist, either trim the ship differently or increase

the metacentric height GM.

It should also be pointed out that parametric excited

roll motion only occurs when the wave period of

encounter is about half the roll resonance period. Hence, if parametrically excited roll motion is experi-enced at sea, which is most likely to be expected in following wave conditions, a change of course to a

more beam or head sea condition will reduce the

problem.

Finally, it should be emphasized that forward veloc-ity will increase the roll damping, sometimes by more than a factor of two. Roll problems experienced at zero speed may thus disappear at a forward velocity. This is believed to have been the case experienced by a tanker loading offshore. Hooked up to the moor-ing system the ship experienced roll angles of _{up to} 20 degrees in a head sea condition. A change in head-ing (reduction of the relative motion) eliminated the problem, a problem never experienced when operated on route from port to port.

11. CONCLUSION

; In this paper it has been demonstrated that a ship

(19)

14 15 16 17 1 6

9-13

Fig. 26. Underwater body of the model used in the experiments.

model in a random seaway may be parametrically excited into roll motion if the modal frequency of the encountered wave spectrum is close to the double of the ships natural frequency of roll.

The effect is most significant in following seas, but may also occur in beam seas, increasing the normally experienced roll angles.

The phenomenon is only encountered by ships with, a large change in the water plane areawhen moving in waves. In addition the roll damping must be small which is often the case on ships operating without bilge keels.

However, a large change in the water plane area with draught will not always give rise to parametric

excited roll motion. If the Mathieu parameter is

strongly non-harmonic and not monotonously

increasing, it

is most likely to give a stable roll

motion.

As discussed in this paper there are a number of factors which are of importance to parametricexcited

roll motion. These are shortly summarized

in the

following:

A small GM increases the risk of

experiencing

parametric excited roll motion.

A long natural period of roll (18-26 secs) increases the probability of experiencing parametric excited roll motion. This being the case since the relative vertical motion of the ship, a measure on the energy transferred into roll, is most stimulated in seaways with a wave period of 9-13 secs. (About half that of the natural roll period).

Increased roll damping decreases the probability of having parametric excited roll motion. Increased damping may be obtained from increased forward velocity or increased area of the bilge keels. The wave frequency of encounter should be double that of the natural period of roll to have a para-metrically excited roll motion. By changing the ship's heading angle (or speed) and hence the fre-quency of encounter the parametric excited roll motion will be eliminated.

NOMENCLATURE

aWL Area of waterplane [m21

A Added moment of inertia in roll [kgrnI

BL BBK Bc GM0 LBK TZ Zrel ZwL Psw we COo

'5

6-8

Continued on page 46

Beam, moulded _Eml

Linear damping coefficient in roll [kgm'fsi

Breadth of bilge keel _Eml

Cubic damping coefficient in roll, non-dimensional

Linear damping coefficient in roll

Restoring coefficient in roll [Nmi

draught

Exciting moment in roll [Nm]

gravity

_[n/s9

Initial metacentric height _[mi

Mathieu parameter

Moment of inertia in roll [kgin2i

Length between perpendiculars Eml

Length of bilge keel [ml

Zero crossing period (s1

Vertical relative motion Erni

Distance from waterline to centre [m] of gravity

Ordinate for centre of flotation Displacement

Initial displacement al Tuning factor in roll Mathieu parameter

Roll angle, -velocity and -acceler-ation [rad, rad/s, rad/e] Wave amplitude

Non-dimensional cubic damping

coefficient

Non-dimensional linear damping

coefficient

Density of seawater Wave frequency

Wave frequency of encounter Natural frequency of roll

BL 20)o(I + A) [m] [kg] [kg/m3J [radis I [rad/4 [radis I

(20)

rirContinued

from page 20 REFERENCES

Salvesen, N., Tuck, E. 0. and Faltinsen, 0.: «Ship Motions and Sea Loads». Trans. Sname, Vol. 78,

1970.

Grim, 0.: «Rollschwingungen, Stabilitat und Sich-erheit im Seegang». Schiffstechnik, 1952, 1,10.

Kerwin, J. E.: «Notes on rolling i longitudinal Waves». Int. Shipb. Prog. 1955, 2, 597.

Paulling, J. R., Rosenberg, R. M.: «On Unstable Ship Motions Resulting from Non-linear Coup-ling». Journal of Ship Res. June 1959.

Abicht, W.: «On Capsizing of Ships in Regular and Irregular Seas». Proc. The Int. Conf. on Sta-bility of Ships and Ocean Vehicles, March 1975, Univ. of Strathclyde, Glasgow, Scotland.

Bang, C. J., Kure, K.: «Hurtige cargoliners Ruin-ing». Hydro-og Aerodynamisk lab. Danmark Jan.

1971.

Blocld, W.: «Ship safety in connection with par-ametric resonance of the roll». Int. Shipb. Progr. Feb. 1980, Vol. 27.

Feat, G. and Jones, D.: «Parametric excitation and the stability of a ship subjected to a steady heeling moment». Int. Shipb. Progr. Nov. 1981, Vol. 28.

Haddara, Kastner, Magel, Paulling, Perezy Perez, Wood: «Capsizing experiments with a model of a fast cargoliner in San Francisco bay». Final

report. Jan. 1972. DOT-CG-84, 549-A. U.S. Coast

Guard. Dept. of Naval Arch. Univ. of Calif.

Berkeley.

Price, W. G.: «A Stability Analysis of the Roll Motion of a Ship in an Irregular Seaway». Int. Shipb. Progr. Vol. 22, pp. 103-112, 1975.

Haddara, M. R.: «A Study on the Stability of the Mean and Variance of Rolling Motion in Random Waves». Int. Conf. on Stability of Ships and Oc. Vehicles. March 1975. Univ. of Strathclyde, Glas-gow, Scotland.

Muhuri, P. K.: «A Study of the Stability of the Rolling Motion of a Ship in an Irregular Seaway». Int. Shipb. Progr. Vol. 27, pp. 1397142, 1980. Roberts, J. B.: «The Effect of Parametric

Exci-tation on Ship Rolling Motion in Random Waves».

National Maritime Institute, NMI R 100, October

1980.

Stoker, J. J.: «Non-linear vibrations in mechanical and electrical systems». Interscience 1950. Skomedal, N.: «Parametric Excitation of Roll Motions and Its Influence on Stability». The Sec-ond Int. Conf. on Stability of Ships and Ocean Vehicles, Tokyo 1982.

Dalzell, J. F.: «A Note on the Form of Ship Roll Damping». Joum. of Ship Res. Vol. 22, No. 3,

1978.

L

46

17. Krogh, F.: «Supersim, an integrated program sys... tern for simulation of dynamic systems*. VERL. TAS Report No. 75-92-C, 1975.

00.00. COCO COO COO. OEM

Continued from page 23

1000 800 $7 600 400 le; ci 200

by corroding steel in concrete failures and signals caused by other nearby corroding steels.

The method should be a valuable supplement to visual inspection on concrete structures under water.

REFERENCES

Browne, R. D. et al.: «Inspection and Monitoring of Concrete Structures for Steel Corrosion*. 9th Offshore Technology Conference, May 2-5 1977. Houston.

Eggen, T. G., Strommen, R., Nilsen N., and Bar-dal, E.: «Nytt maleutstyr og mileprinsipp for kon-troll av katodisk vem». Norsk Oljerevy Bd. 4 fl(.

8, 1978.

Eggen, T. G.: «Anvendelse av T-sensor pa betong-konstruksjonem SINTEF Report STF65 A81065. NTH 1981.

0000000000000Ceeee

20 40 60 80 100

Distance from crack (mm)

Fig. 8. Sensor signals with varying distance from